The integration notes discuss how to properly patch Chai with
the plugin in different environments (browser, AMD, Node.js). The
API documentation is a good starting point to learn how to use
chai-jq assertions in your tests.
The site's own test suites also provide a decent introduction to the API:

Asserts that the target has exactly the given CSS property, or
asserts the target contains a subset of the CSS when using the
include or contain modifiers.

Node.js/JsDom Note: Computed CSS properties are not correctly
inferred as of JsDom v0.8.8. Explicit ones should get matched exactly.

Browser Note: Explicit CSS properties are sometimes not matched
(in contrast to Node.js), so the plugin performs an extra check against
explicit style properties for a match. May still have other wonky
corner cases.

PhantomJS Note: PhantomJS also is fairly wonky and unpredictable with
respect to CSS / styles, especially those that come from CSS classes
and not explicity style attributes.